Topic: Werckmeister's Septinarius temperament
5 scales
| File | Description | Notes | Period (¢) | Limit |
|---|---|---|---|---|
| sep | Septanarius scale? | 12 | 1200.0 | 139 |
| septenarius | Septenarius scale ('Werckmeister VI') | 12 | 1200.0 | 139 |
| septenarius440Hzmk2 | TD's septenarius @ middle c'=262Hz or a'=440Hz | 12 | 1200.0 | 139 |
| septenarius_GG49Hz | sparschuh's version @ middle-c'=262Hz or a'=440Hz | 12 | 1200.0 | 131 |
| septenarius_tuning_69724_69750 | Septenarius scale ('Werckmeister VI') | 12 | 1200.0 | 139 |
Thread (26 messages)
From: Gene Ward Smith (2007-02-12) Subject: Werckmeister's Septinarius temperament Tom has a web page on this I'm trying to decipher: http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html It it he presents the deviations from JI fifths for his two reconstructions of this temperament in terms of a new (to me) and horrible notation for positive rational numbers, whereby p/q > 1 is written +p:q and p/q < 1 is written -p:q. Could we PLEASE stick to the standard mathematical notation everyone learned in grade school? Now that I've vented about that (sorry Tom, but you hit my pet peeve button) I'll give the deviations for "D=175", starting from C-G and working the circle of fifths around to F-C: 392/393, 524/525, 350/351, 1, 1, 416/417, 1, 1, 1, 1, 440/441, 1 Now, a problem with this is that the prodcut of this deviations isn't 524288/531441, so octaves are slightly tempered. Presuming this isn't intentional, at least one of these ratios is off, and in fact G#-D# is given as 496/496. If instead we make that 4448/4455, we get untempered fifths. In the vague hope that this is more or less what is intented, here's a first go at what this temperament would be: ! sep.scl Septanarius scale? 12 ! 1568/1485 28/25 196/165 49/39 4/3 196/139 196/131 784/495 196/117 98/55 49/26 2
From: Carl Lumma (2007-02-12) Subject: Re: Werckmeister's Septinarius temperament > Could we PLEASE stick to the > standard mathematical notation everyone learned in grade school? // > here's a first go at what this temperament would be: > > ! sep.scl > Septanarius scale? > 12 > ! > 1568/1485 > 28/25 > 196/165 > 49/39 > 4/3 > 196/139 > 196/131 > 784/495 > 196/117 > 98/55 > 49/26 > 2 Thank you, Gene. -Carl
From: Mohajeri Shahin (2007-02-13) Subject: RE: [tuning] Werckmeister's Septinarius temperament Hi gene your scale has degrees of 1568-edl you can see something about this EDl-based well temperament and 196-EDL and The Septenarius, Werckmeister's mythical tuning in : http://240edo.googlepages.com/equaldivisionsoflength(edl) it is interesting that 1568=196*7. Shaahin Mohajeri Tombak Player & Researcher , Microtonal Composer My web site?? ???? ????? ?????? <http://240edo.googlepages.com/> My farsi page in Harmonytalk ???? ??????? ?? ??????? ??? <http://www.harmonytalk.com/mohajeri> Shaahin Mohajeri in Wikipedia ????? ?????? ??????? ??????? ???? ???? <http://en.wikipedia.org/wiki/Shaahin_mohajeri> ________________________________ From: [email protected] [mailto:[email protected]] On Behalf Of Gene Ward Smith Sent: Monday, February 12, 2007 11:47 PM To: [email protected] Subject: [tuning] Werckmeister's Septinarius temperament Tom has a web page on this I'm trying to decipher: http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html <http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html> It it he presents the deviations from JI fifths for his two reconstructions of this temperament in terms of a new (to me) and horrible notation for positive rational numbers, whereby p/q > 1 is written +p:q and p/q < 1 is written -p:q. Could we PLEASE stick to the standard mathematical notation everyone learned in grade school? Now that I've vented about that (sorry Tom, but you hit my pet peeve button) I'll give the deviations for "D=175", starting from C-G and working the circle of fifths around to F-C: 392/393, 524/525, 350/351, 1, 1, 416/417, 1, 1, 1, 1, 440/441, 1 Now, a problem with this is that the prodcut of this deviations isn't 524288/531441, so octaves are slightly tempered. Presuming this isn't intentional, at least one of these ratios is off, and in fact G#-D# is given as 496/496. If instead we make that 4448/4455, we get untempered fifths. In the vague hope that this is more or less what is intented, here's a first go at what this temperament would be: ! sep.scl Septanarius scale? 12 ! 1568/1485 28/25 196/165 49/39 4/3 196/139 196/131 784/495 196/117 98/55 49/26 2
From: Tom Dent (2007-02-13) Subject: Re: Werckmeister's Septinarius temperament --- In [email protected], "Carl Lumma" <clumma@...> wrote: > > > Could we PLEASE stick to the > > standard mathematical notation everyone learned in grade school? > // > > here's a first go at what this temperament would be: > > > > ! sep.scl > > Septanarius scale? > > 12 > > ! > > 1568/1485 > > 28/25 > > 196/165 > > 49/39 > > 4/3 > > 196/139 > > 196/131 > > 784/495 > > 196/117 > > 98/55 > > 49/26 > > 2 > > Thank you, Gene. > > -Carl > ... Once more - this scale is NOT correct. Honestly, all you have to do is read the webpage from start to finish to find out exactly what Werckmeister said. I have no idea why people are unwilling to do this. ~~~T~~~
From: Tom Dent (2007-02-13) Subject: Werckmeister's Septenarius temperament --- In [email protected], "Gene Ward Smith" <genewardsmith@...> wrote: > > Tom has a web page on this I'm trying to decipher: > > http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html > > It it he presents the deviations from JI fifths for his two > reconstructions of this temperament in terms of a new (to me) and > horrible notation for positive rational numbers, whereby p/q > 1 is > written +p:q and p/q < 1 is written -p:q. Could we PLEASE stick to the > standard mathematical notation everyone learned in grade school? > The DEFINITION of the scale is in the monochord numbers, which are the first table in the webpage. I was working on the assumption that people would start at the beginning and read towards the end... Therefore the scale is defined to be (apologies for malformed Scala) ! sep.scl Septenarius scale (choose either value of D) 12 ! 1 196/186 = 98/93 196/176 or 196/175 196/165 196/156 = 49/39 196/147 = 4/3 196/139 196/131 196/124 = 49/31 196/117 196/110 = 98/55 196/104 = 49/26 The notation Gene dislikes is not a notation for numbers; it is a notation for tempering of fifths. It's actually the way Werckmeister set out his fifths. It's certainly not the definition of the tuning. Anyway, you are correct that G#-D# is a typo on my part. Try a wide fifth tempered by 496/495. Where Gene got 4448/4455 from I can't tell. 392/393 * 524/525 * 350/351 * 416/417 * 278/279 * 496/495 * 440/441 *(3^12)/(2^19) = 1 No need for any integer exceeding 525! ~~~T~~~
From: Cameron Bobro (2007-02-13) Subject: Re: Werckmeister's Septenarius temperament --- In [email protected], "Tom Dent" <stringph@...> wrote: > > --- In [email protected], "Gene Ward Smith" <genewardsmith@> > wrote: > > > > Tom has a web page on this I'm trying to decipher: > > > > http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html > > > > It it he presents the deviations from JI fifths for his two > > reconstructions of this temperament in terms of a new (to me) and > > horrible notation for positive rational numbers, whereby p/q > 1 is > > written +p:q and p/q < 1 is written -p:q. Could we PLEASE stick to the > > standard mathematical notation everyone learned in grade school? > > > > The DEFINITION of the scale is in the monochord numbers, which are the > first table in the webpage. I was working on the assumption that > people would start at the beginning and read towards the end... > > > Therefore the scale is defined to be (apologies for malformed Scala) > > ! sep.scl > Septenarius scale (choose either value of D) > 12 > ! > 1 > 196/186 = 98/93 > 196/176 or 196/175 > 196/165 > 196/156 = 49/39 > 196/147 = 4/3 > 196/139 > 196/131 > 196/124 = 49/31 > 196/117 > 196/110 = 98/55 > 196/104 = 49/26 > > > The notation Gene dislikes is not a notation for numbers; it is a > notation for tempering of fifths. It's actually the way Werckmeister > set out his fifths. It's certainly not the definition of the tuning. > > Anyway, you are correct that G#-D# is a typo on my part. Try a wide > fifth tempered by 496/495. Where Gene got 4448/4455 from I can't tell. > > 392/393 * 524/525 * 350/351 * 416/417 * 278/279 * 496/495 * 440/441 > *(3^12)/(2^19) = 1 > > No need for any integer exceeding 525! > > ~~~T~~~ Apparently you missed this earlier post of mine... > 0: 1/1 0.000 unison, perfect prime > 1: 98/93 90.661 > 2: 28/25 196.198 middle second > 3: 196/165 298.065 > 4: 49/39 395.169 > 5: 4/3 498.045 perfect fourth > 6: 196/139 594.923 > 7: 196/131 697.544 > 8: 49/31 792.616 > 9: 196/117 893.214 > 10: 98/55 1000.020 quasi-equal minor seventh > 11: 49/26 1097.124 > 12: 2/1 1200.000 octave > > EDL is as old as the hills, isn't it? > > Cents schments, I'm digging the ratios, > > Hmmm....sounds pretty damn good! > > Thanks, Tom Dent!
From: Carl Lumma (2007-02-13)
Subject: Re: Werckmeister's Septenarius temperament
> Therefore the scale is defined to be (apologies
> for malformed Scala)
>
> ! sep.scl
> Septenarius scale (choose either value of D)
> 12
> !
> 1
> 196/186 = 98/93
> 196/176 or 196/175
> 196/165
> 196/156 = 49/39
> 196/147 = 4/3
> 196/139
> 196/131
> 196/124 = 49/31
> 196/117
> 196/110 = 98/55
> 196/104 = 49/26
Believe it or not, Tom, I have about 2323 phone calls, 30949
e-mails, and 3909409 web pages to answer and read every day,
only a fraction of them about music. So when sharing a share
with this community, people like me really appreciate that
you use the standard way of communicating scales here, which
is giving a Scala file as you've done (finally). You can make
it well-formed simply by giving the period instead of the
unity, and by prefixing your comments on each line with a
bang, like this:
! septenarius.scl
Septenarius scale ('Werckmeister VI')
12
!
98/93 ! =196/186
196/176 ! or 196/175
196/165
49/39 ! =196/156
4/3 ! =196/147
196/139
196/131
49/31 ! =196/124
196/117
98/55 ! =196/110
49/26 ! =196/104
2
!
Now I'm going to scan through the boring historical
stuff on your page for an explanation as to why there
is a choice involving the second degree. Anything you
can do to speed that up would also be appreciated.
-Carl
From: Carl Lumma (2007-02-13)
Subject: Re: Werckmeister's Septenarius temperament
> Now I'm going to scan through the boring historical
> stuff on your page for an explanation as to why there
> is a choice involving the second degree. Anything you
> can do to speed that up would also be appreciated.
! septenarius.scl
Septenarius scale ('Werckmeister VI')
12
!
98/93 ! =196/186
196/175 ! 196/176 apparently a typo
196/165
49/39 ! =196/156
4/3 ! =196/147
196/139
196/131
49/31 ! =196/124
196/117
98/55 ! =196/110
49/26 ! =196/104
2
!
-Carl
From: Carl Lumma (2007-02-13)
Subject: Re: Werckmeister's Septenarius temperament
> ! septenarius.scl
> Septenarius scale ('Werckmeister VI')
> 12
> !
> 98/93 ! =196/186
> 196/175 ! 196/176 apparently a typo
> 196/165
> 49/39 ! =196/156
> 4/3 ! =196/147
> 196/139
> 196/131
> 49/31 ! =196/124
> 196/117
> 98/55 ! =196/110
> 49/26 ! =196/104
> 2
> !
It's got one sharp fifth, but no thirds sharper than
81/64 like Paul's Continuo tuning. It does have a fifth
even flatter than the 1/4-comma fifths he so decries in
that text.
-Carl
-Carl
From: Tom Dent (2007-02-13)
Subject: Re: Werckmeister's Septenarius temperament
--- In [email protected], "Carl Lumma" <clumma@...> wrote:
>
> > ! septenarius.scl
> > Septenarius scale ('Werckmeister VI')
> ...
>
> It's got one sharp fifth, but no thirds sharper than
> 81/64 like Paul's Continuo tuning. It does have a fifth
> even flatter than the 1/4-comma fifths he so decries in
> that text.
>
> -Carl
Indeed...
"the major thirds are perfectly consistent with Werckmeister's other
tunings [of 1691]: the purest lie at F-A and C-E, while the
little-used thirds at F#-Bb, C#-F and G#-C are all nearly a comma
sharp. The note D#, which must also do duty as Eb, is placed almost
equally between B (natural) and G, approaching Equal Temperament in
this triple of major thirds."
I think the 'Septenarius' is to be compared with the other organ
tunings of 1691 - including 'IV' which has 1/3 comma flat fifths (and
some thirds 1/3 comma wider than 81/64).
But by 1698, and discussing stringed keyboard instruments,
Werckmeister seems to have been singing a different tune.
It seems to me one has to decide whether or not one is interested in a
subject enough to spend time on it. If you're not really interested in
historical tuning, I think it's unrealistic to hope to understand
anything with five minutes' skimming. It's complicated and subtle and
can't be served up on a plate.
I decided not to include any cent values or fractions of a comma, if
at all possible, because that was not how Werckmeister conceived of
this tuning. (Logarithmic approaches have in the past led to incorrect
versions of it.) To understand what it is about - and 196-EDL is more
or less correct in that sense - you have to look at how it was worked
out. There is no short cut. It is possible to find out anything you
need to know about it with a bit of integer arithmetic.
~~~T~~~
From: Gene Ward Smith (2007-02-13) Subject: Re: Werckmeister's Septinarius temperament --- In [email protected], "Tom Dent" <stringph@...> wrote: > ... Once more - this scale is NOT correct. Honestly, all you have to > do is read the webpage from start to finish to find out exactly what > Werckmeister said. I have no idea why people are unwilling to do this. Give the scale or don't give it, but please don't blame other people for your failures.
From: Gene Ward Smith (2007-02-13) Subject: Re: Werckmeister's Septenarius temperament --- In [email protected], "Tom Dent" <stringph@...> wrote: > The notation Gene dislikes is not a notation for numbers; it is a > notation for tempering of fifths. Which are NUMBERS. This is, without doubt or question, a nonstandard notation for positive rational numbers. Had they been given correctly in this notation or a standard one I could have derived the scale from them, as I tried to do. I liked that idea better, since I was, I thought, on firm ground with a circle of fifths, whereas it wasn't as clear to me what you were saying about the monochord. It sounded as if it might be some kind of starting point and not final product, and when I noticed the circle of fifths didn't come to an exact octave, that raised more concerns. > It's actually the way Werckmeister > set out his fifths. It's certainly not the definition of the tuning. If you want to define a scale of rational numbers most perspicuously, give it as an ascending sequence of rational numbers. However, a circle of fifths will in fact define the tuning. And if Werckmeister uses a clumsy, nonstandard notation, do not follow him. Make a note of it if you like, but don't *adopt* it. What's commonly done in history of math or science when someone's work is discussed is that the notation is modernized so that the substance can be seen more clearly.
From: Gene Ward Smith (2007-02-13) Subject: Re: Werckmeister's Septenarius temperament --- In [email protected], "Carl Lumma" <clumma@...> wrote: > Now I'm going to scan through the boring historical > stuff on your page for an explanation as to why there > is a choice involving the second degree. Anything you > can do to speed that up would also be appreciated. The alternative is Werckmeister's actual figures, but those give screwy results, and assuming he made a writeo clears that up.
From: Tom Dent (2007-02-14) Subject: Re: Werckmeister's Septinarius temperament --- In [email protected], "Gene Ward Smith" <genewardsmith@...> wrote: > > --- In [email protected], "Tom Dent" <stringph@> wrote: > > > ... Once more - this scale is NOT correct. Honestly, all you have to > > do is read the webpage from start to finish to find out exactly what > > Werckmeister said. I have no idea why people are unwilling to do this. > > Give the scale or don't give it, but please don't blame other people > for your failures. MY failures? What the hell do you mean by that? The very first table in the webpage gives the scale with complete clarity; for an unfathomable reason, some people ignored it. I made ONE easily identifiable typo in a subsequent table: 496:496 instead of 496:495. The correct ratio could be deduced using the table of monochord numbers and some elementary arithmetic. For some reason some people managed to screw that up and get a nonsensical ratio of 4-figure integers instead. I'm not responsible for that. How about we talk about your failures too? And everyone else's failures? What a NICE discussion that would make. ~~~T~~~
From: Carl Lumma (2007-02-14) Subject: Re: Werckmeister's Septenarius temperament > It seems to me one has to decide whether or not one is > interested in a subject enough to spend time on it. If you're > not really interested in historical tuning, I think it's > unrealistic to hope to understand anything with five minutes' > skimming. It's complicated and subtle and can't be served up > on a plate. The dynamic range of WTs is quite narrow, even if you do things like allow sharp fifths. Listening tests here suggest that it's harder to distinguish them than theoretical discourse implies. The music sounds just fine in any number of WTs. I personally don't like any thirds over about 404 cents in keys I plan to use. So this tuning isn't of much interest to me. I'd never use a tuning because it's "historical", because as I say, I don't think we have the records of high enough quality to make that term very meaningful. Case in point, there's a serious typo/error in the present text, and this doesn't seem unusual for these texts. The study of tunings is today a niche, and clearly it was even less of one in the baroque or there'd be more/better records. > I decided not to include any cent values or fractions of a > comma, if at all possible, because that was not how > Werckmeister conceived of this tuning. People of antiquity did all kinds of crazy things, like wear wigs. Do you put one on before listening? -Carl
From: Paul Poletti (2007-02-14) Subject: Git real! Carl Lumma wrote: > > I personally don't like any thirds over about 404 > cents in keys I plan to use. So this tuning isn't of much > interest to me. I'd never use a tuning because it's > "historical", because as I say, I don't think we have the > records of high enough quality to make that term very > meaningful. Case in point, there's a serious typo/error > in the present text, and this doesn't seem unusual for > these texts. The study of tunings is today a niche, and > clearly it was even less of one in the baroque or there'd > be more/better records. > Oh get off it, Carl! There are a lot of serious people doing a lot of very good and solid research in this field. There's also a lot of good, solid evidence about what was done in the past. If you don't like the results, then just say you don't like it; don't go inventing problems which don't exist. You sound like a Bush administration lackey talking about global warming, or a cigarette manufacturer claiming there's no link to lung cancer. If you want to know just how incredibly uninformed you seem by making the above statement, first go learn German, then read at least Mark Lindley's Stimmung und Temperatur, if not the original sources themselves. Ciao, Paul
From: Tom Dent (2007-02-16) Subject: Re: Werckmeister's Septenarius temperament --- In [email protected], "Carl Lumma" <clumma@...> wrote: > > Listening tests here suggest > that it's harder to distinguish [WTs] than theoretical > discourse implies. That's mainly because most of the ones that have been tested here aren't very unequal, and/or because of acoustic conditions (timbre and sustain). Try temperament ordinaire with a bright pipe organ timbre and you'll notice pretty soon. Or Kirnberger II ... not that I'm recommending it, just that it should be obvious. > I don't think we have the > records of high enough quality to make that term very > meaningful. This particular Werckmeister effort is a 'worst case' scenario, in that he chose an innovative way of constructing a temperament and slipped up a bit in labeling his monochord. Almost all historical sources that people take seriously are a lot *less* problematic than the 'Septenarius'... But you won't discover that if you don't look at any. > People of antiquity did all kinds of crazy things, like > wear wigs. ... and use just intonation and microtonal scales. How can we possibly distinguish between wig-wearing and unequal tuning? It's all Stuff of Antiquity, innit? ~~~T~~~
From: Gene Ward Smith (2007-02-16) Subject: Re: Werckmeister's Septenarius temperament --- In [email protected], "Tom Dent" <stringph@...> wrote: > That's mainly because most of the ones that have been tested here > aren't very unequal, and/or because of acoustic conditions (timbre and > sustain). Try temperament ordinaire with a bright pipe organ timbre > and you'll notice pretty soon. Or Kirnberger II ... not that I'm > recommending it, just that it should be obvious. When in the history of circulating temperament did anyone *ever* use a temperament as unequal as grail--which tested out very nicely?
From: Tom Dent (2007-02-17) Subject: Jean-Philippe Rameau and the Holy Grail --- In [email protected], "Gene Ward Smith" <genewardsmith@...> wrote: > > --- In [email protected], "Tom Dent" <stringph@> wrote: > > > That's mainly because most of the ones that have been tested here > > aren't very unequal, and/or because of acoustic conditions (timbre and > > sustain). Try temperament ordinaire with a bright pipe organ timbre > > and you'll notice pretty soon. Or Kirnberger II ... not that I'm > > recommending it, just that it should be obvious. > > When in the history of circulating temperament did anyone *ever* use a > temperament as unequal as grail--which tested out very nicely? > Rameau's 1726 advice asks for a run of 7 quarter-comma fifths (starting on either C or Bb) then gradual widening over the remaining 5, the last two being probably the widest to reach the starting-point again. If we start on Bb then Bb-D is pure and D-F# is nearly pure; therefore F#-Bb is likely to be about 423 cents. Indeed, Grail is a lot more unequal than that. I rather suspect Grail sounded good because the piece fitted it... What about putting together a piece in B minor, which (among familiar keys) should be the reverse of G minor? BWV544? If one wanted to make a synthesized comparison to Rameau I would take his tuning starting from C, with C#-G# pure and the remaining four fifths sharing the leftover wideness. This will be (7*3 - 13)/12 = 2/3 comma, so the four sharp fifths can have 1/6 comma each. ~~~T~~~
From: Cameron Bobro (2007-02-18) Subject: Re: Werckmeister's Septenarius temperament --- In [email protected], "Carl Lumma" <clumma@...> wrote: > Listening tests here suggest > that it's harder to distinguish them than theoretical > discourse implies. If we were to have a color test with things like cold orange-reds and a warm red-oranges, would you say that the colors are hard to distinguish based on a general failure of the testees to name the correct Photoshop hex-codes? We don't really know how differently the WTs sounded to the testees: there were several cases where people noted that they could hardly tell the difference between x and y (somewhere there were two that I couldn't distinguish at all, for example) but otherwise, most of the responses described specific impressions. And these tests were often handicapped by questionable Soundfont tuning accuracy. The tests showed only that in these conditions, the testees couldn't name the specific WT scheme consistently. Of course, you could say that the testees were simply imagining differences. This could only be tested by sneaking in doubles, and even then, just because a person imagines a difference in the case of two identical examples doesn't mean that he's imagining differences between things that actually are different. Not to mention the fundamental difference between hearing things in the very emotional experience of listening to music and the this clinical, repetitive (and probably even stressful for some) experience, and you just can't make such a general statement based on these informal tests. -Cameron Bobro
From: Carl Lumma (2007-02-18) Subject: Re: Werckmeister's Septenarius temperament > > Listening tests here suggest > > that it's harder to distinguish them than theoretical > > discourse implies. > > If we were to have a color test with things like cold orange-reds > and a warm red-oranges, would you say that the colors are hard to > distinguish based on a general failure of the testees to name the > correct Photoshop hex-codes? No, but I couldn't even tell the difference between all of the scales. And as I said, I wasn't trying to identify them, either; just rate their consonance and match this up with the name of the temperament with the most consonant most consonant key. Without extensive listening to all of the scales in all keys, no testee could hope to identify them, and few claimed to be trying. > And these tests were often > handicapped by questionable Soundfont tuning accuracy. Not questionable at all. > Not to mention the fundamental difference between hearing things in > the very emotional experience of listening to music and the this > clinical, repetitive (and probably even stressful for some) Yes, all of this has been mentioned by the testees. > experience, and you just can't make such a general statement based > on these informal tests. None of this applies to the soundness of my statement. Maybe try reading it again (at the top). -Carl
From: Andreas Sparschuh (2007-04-12) Subject: New septenarius for a'=440 Hz, was Re: Werckmeister's Septinarius temperament --- In [email protected], "Gene Ward Smith" <genewardsmith@...> wrote/discussed in: > >http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html W's modified stringlengths can be interpreted as absolute frequencies of an 5ths circle: a'440Hz a220 A110 e'330 e165 b' 495 1485/1484 f#"742 f#'371 1113/1112 c#"556 c#'278 c#139 417/416 g#208 G#104 eb'312 eb156 Eb78 bb'468 bb234 Bb117 f'351 1053/1052 c"526 c'263/262 131 g'393/392 196 98 49=7*7 d147 441/a'440 cps on the keys: _____________________ | c' 263 middle-C +---------|278 c#'=db' | d' 294 +---------|294 eb'=d#' | e' 330 +--------------------- | f' 351 +---------|371 f#'=gb' | g' 393 +---------|416 g#'=ab' | a' 440 Hz +---------|468 bb'=a#' | b' 468 +-------------------- | c" 526 +---------| c#" &ct... !septenarius440Hz.scl ! sparschuh's septenarius @ middle c'=263Hz or a'=440Hz ! 12 ! 278/263 ! C# 294/263 ! D 312/263 ! Eb 330/263 ! E 351/263 ! F 351/263 ! F# 393/263 ! G 416/263 ! G# 440/263 ! A 468/263 ! Bb 495/263 ! B 2/1 Just try it out to play in that yourself! http://www.strukturbildung.de/Andreas.Sparschuh
From: Gene Ward Smith (2007-04-12) Subject: New septenarius for a'=440 Hz, was Re: Werckmeister's Septinarius temperament --- In [email protected], "Andreas Sparschuh" <a_sparschuh@...> wrote: > !septenarius440Hz.scl > ! > sparschuh's septenarius @ middle c'=263Hz or a'=440Hz > ! > 12 > ! > 278/263 ! C# > 294/263 ! D > 312/263 ! Eb > 330/263 ! E > 351/263 ! F > 351/263 ! F# > 393/263 ! G > 416/263 ! G# > 440/263 ! A > 468/263 ! Bb > 495/263 ! B > 2/1 F and F# are the same note.
From: Tom Dent (2007-04-13) Subject: New septenarius for a'=440 Hz, was Re: Werckmeister's Septinarius temperament Comments below! --- In [email protected], "Andreas Sparschuh" <a_sparschuh@...> wrote: > > >http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html > > W's modified stringlengths can be interpreted as absolute frequencies > of an 5ths circle: > > a'440Hz a220 A110 > e'330 e165 > b'495 > 1485/1484 f#"742 f#'371 Instead of 496 and 372. This avoids having a wide e-b fifth and is better for G major and D major. > 1113/1112 c#"556 c#'278 c#139 > 417/416 g#208 G#104 > eb'312 eb156 Eb78 > bb'468 bb234 Bb117 > f'351 > 1053/1052 c"526 c'263/262 131 > g'393/392 196 98 49=7*7 I don't think this is so good, you have C-G tempered by 262/263 which is about 1/3 comma... Better to have 315/350 f'350 f175 and 525/524 c''524 c262 131 ... ? Then C-G is pure and > d147 rather 1179/1176 d''588 d'294 > 441/a'440 cps > > on the keys: [edited] > _____________________ > | c' [262] middle-C > +---------|278 c#'=db' > | d' 294 > +---------|[312] eb'=d#' > | e' 330 > +--------------------- > | f' [350] > +---------|371 f#'=gb' > | g' 393 > +---------|416 g#'=ab' > | a' 440 Hz > +---------|468 bb'=a#' > | b' [495] > +-------------------- > | c" [524] > +---------| c#" > &ct... > > !septenarius440Hz.scl > ! > sparschuh's septenarius @ middle c'=263Hz or a'=440Hz > ! > 12 > ! > 278/263 ! C# > 294/263 ! D > 312/263 ! Eb > 330/263 ! E > 351/263 ! F > 351/263 ! F# [should be 271] > 393/263 ! G > 416/263 ! G# > 440/263 ! A > 468/263 ! Bb > 495/263 ! B > 2/1 > or: !septenarius440Hzmk2.scl ! TD's septenarius @ middle c'=262Hz or a'=440Hz ! 12 ! 278/262 ! C# 294/262 ! D 312/262 ! Eb 330/262 ! E 350/262 ! F 371/262 ! F# 393/262 ! G 416/262 ! G# 440/262 ! A 468/262 ! Bb 495/262 ! B 2/1 (cf. 12ET frequencies: 261.6, 277.2, 293.7, 311.1, 329.6, 349.2, 370.0, 392.0, 415.3, 440.0, 466.2, 493.9 ...) Can one use the nice ratio 63/50 = 1.26 to build a 'septenarian' near- equal tuning? eg Eb-G = 150:189 ... via Eb150 Bb225 (675) F337 (1011) C505 (504) G378=189 then continue: ... (567) D283 (849) A424=212 (636) E635 (634) B951 (2853/2848) F#356=178 C#267 (801) G#400 Eb300 seems to work nicely at late Baroque pitch levels - only three pure fifths between Eb-Bb, G#-Eb, F#-C#. ~~~T~~~
From: Andreas Sparschuh (2007-04-13) Subject: New septenarius for a'=440 Hz, was Re: Werckmeister's Septinarius temperament --- In [email protected], "Tom Dent" <stringph@...> wrote: > > > Comments below! > > > a'440Hz a220 A110 > > e'330 e165 > > b'495 > > 1485/1484 f#"742 f#'371 > > Instead of 496 and 372. This avoids having a wide e-b fifth and is > better for G major and D major. right, in order to get rid of the oversharp wolfs in W's original stringlength numbers, alike in his famous #3 the 'quaternarius' has only 4 flattend and 8 pure 5ths: A E B>F# C# G# Eb Bb F C>G>D>A > > > 1113/1112 c#"556 c#'278 c#139 > > 417/416 g#208 G#104 > > eb'312 eb156 Eb78 > > bb'468 bb234 Bb117 > > f'351 > > 1053/1052 c"526 c'263/262 131 > > g'393/392 196 98 49=7*7 > > I don't think this is so good, you have C-G tempered by 262/263 > which is about 1/3 comma... > ...flattend than a pure 5th: 3/2. So far about that strongest detuned 5th C>G. Consider accordingly the belonging 3rd C>E in the major-chord C>E>G: with compareable sharpness: e165 e'330 e"660 e'"1330/1315=5*c"263 shortening by common factor 5 yiels a tempering of C>E about 264/265, so that the C-major chord consists in C(1/1)-E*(265/264)-G*(262/263) or 4:(5*(265/264)):(6*262/263) instead barely 4:5:6 without the idea of tempering 3rds and 5ths in the same range of magintude. Many 'experts' do recommend to sharpen the 3rds about the amount in amplitude alike the corresponding 5ths become flattened, so that the beatings of 3rds and 5ths beat almost the same in reverse directions. Skilled organ-builders use that effect in order to demonstrate that their robust organs survive even such impressive resonances. It appears that already http://en.wikipedia.org/wiki/Arnolt_Schlick knew that old well-known tuning-trick/method in his instructions. > Better to have 315/350 f'350 f175 and 525/524 c''524 c262 131 ... ? Good idea, if you intend to stay nearer to W's original version, when somehow aiming to approximate "ET" however. > > Then C-G is pure and makes that only sense according the above demands when C-E is also pure chosen? > > > Correction of: !septenarius440Hz.scl > > 351/263 ! F# [should be 271] for Gene: 371 is the correct pitch. > or: > > !septenarius440Hzmk2.scl > ! > TD's septenarius @ middle c'=262Hz or a'=440Hz > ! > 12 > ! > 278/262 ! C# short 138/131 > 294/262 ! D short 147/131 > 312/262 ! Eb short 156/131 > 330/262 ! E short 165/131 > 350/262 ! F short 175/131 > 371/262 ! F# > 393/262 ! G > 416/262 ! G# 208/131 > 440/262 ! A 220/131 > 468/262 ! Bb 234/131 > 495/262 ! B > 2/1 The pure middle c' in reference to a'=440Hz becomes in the just case 440Hz*3/5 = 264Hz. Hence i do prefer the nearer 263Hz instead yours lower 262Hz, which appears just a little bit to flat lowered in my personal taste, especially when having just intonation in mind or ear instead the virtual "et". Most professional and skilled tuners do to keep the frequent keys with few accidentials somehow purer than the less used keys with many accidentials, so that the strange keys got more pythagorean 3rds, by purer or even just pure 5ths inbetween them. > > (cf. 12ET frequencies: 261.6, 277.2, 293.7, 311.1, 329.6, 349.2, > 370.0, 392.0, 415.3, 440.0, 466.2, 493.9 ...) those irrational numbers are far to complicated for solving the problem. Who needs the advanced precision of 4 decimal digits in the octave from c' to c"? But if you want to approach "et" whatsoever, then 262 would be the better choice for converging "et" as the above approximation suggest. > > Can one use the nice ratio 63/50 = 1.26 to build a 'septenarian' >near- > equal tuning? I.m.o: the nearer one draws to approach "et", the most frequently used 3rds get to much worse detuned Luckily nobody can tune irrational intervals in practice, so that the worsest case: "et" remains barely a theoretically fiction, excluded from real implementation on a real sounding instrument. Simply try out how well can you reproduce by yours ears: on the one hand: a pure 5th ratio 3:2=1.5 ~702cents and on the other hand: sqrt(2) = 600 Cent "et"-tritous, that's geometrically interpreted: http://mathworld.wolfram.com/PythagorassConstant.html Experimental result: There is no psychoacusitcally evidence for departening the ratio of a 5th 3:2 for the benefit of sqrt(2) ET-tritone. Quoting Herrmann Helmholtz: "The ear prefers simple ratios." > eg Eb-G = 150:189 ... > via Eb150 Bb225 (675) F337 (1011) C505 (504) G378=189 But by that procedure one does also loose to much of the 'Baroque' key-characteristics. http://www.societymusictheory.org/mto/issues/mto.95.1.4/mto.95.1.4.code.html http://de.wikipedia.org/wiki/Tonartencharakter > > then continue: > ... (567) D283 (849) A424=212 (636) E635 (634) B951 (2853/2848) > F#356=178 C#267 (801) G#400 Eb300 > > seems to work nicely at late Baroque pitch levels - only three pure > fifths between Eb-Bb, G#-Eb, F#-C#. > hmm, is it still apt to call that 'baroque'-style? As far as i understood late "Baroque" post-meantone instructions: There i found a tendency to keep the 5ths inbetween the accidentials F#-C#-G#-Eb-Bb purer more pure (within the upper black keys on the piano) than in the ordinary F-C-G-D-A-E-B: that got generally more tempering. In extreme form i do start from my prototype model. The procedure consists in a chain of 11 almost pure 5ths, that contains the JI pitches, but also schismic Pythagorean-enharmonics: Here the chain F-A-C-E-G-B-D are all 3 pure major 4:5:6 chords in exact beatless just proportions: A 440. 220 110 55 E 165 B 495. F# 1485 4455 C# 4454 2227 6681 G# 6680 3340 1670 835 2505 Eb 2504 1252 626 313. Bb 939 2817 F 2816 1408 704 352. 176 88 44 22 11 C 33 G 99 D 297. / 296 148 74 37 111 A 110 55 A E B F#~C#~G#~Eb Bb~F C G D~~~~~~~~~~~~~~~~~~~~~~~~~~~~A The schisma 32805/32768=5*3^8/2^15= (4455/4454)(6681/6680)(2505/2504)(2817/2816) is tempered out by the subdivsion in to that product of 4 superparticular factors. Respectively the SC=81/80=(297/296)(111/110) into 2 parts at one @ D>A 40:27 Rearranging same pitches in ascending order yields: C' 264Hz middle-C C# 278.375 D' 297 Eb 313 E' 330 F' 352 F# 371.25 G' 396 G# 417.5 A' 440Hz reference pitch Bb 469.5 B' 495 C" 528 so far about the 11 other frquencies that i percieve instantly also in mind immediatley when hearing a 440Hz tuning-fork by ear, or simpy when imagening that pitch-levels enwraped when reading musical scores in any fitting tuning. schismatic_just440Hz.scl ! sparschuh's-schisma-subdivision(4455/4454)(6681/6680)(2505/2504)(2817/2816) ! 2227/2112 ! C# 9/8 ! D 313/264 ! Eb 5/4 ! E 4/3 ! F 45/32 ! F# 3/2 ! G 835/528 ! G# 5/3 ! A=440Hz 313/176 ! Bb 15/8 ! B 2/1 But, how about that almost similar alternative one at the moment on my piano? A 440. 220 110 330 E 329. B 987 2961 F# 2960 1480 740 370. 185 C# 555 1665 G# 1664 832 416. 208 104 52 26 13 Eb 39 Bb 117 F 351. 1053 C 1052 526 263. 789 G 788 394. 197 591 D 590 285./284 147 441 A 440. (or 3*285=885 A 880 440.) with strongest tempering @ D>A: 885/880=(285/284)(441/440)=177/176 but still less than SC^(1/2) ~161/160 or ~162/161, hence rather tolerable than the ancient Erlangen-monochord or Kirnberger#1, that charge a full SC on D>A alike the above 'schismatic_just.scl'. So far my reccomendation for those who prefer to stay nearer at JI than to the i.m.o. over-detuned "ET", that i do meanwhile consider as outdated intuneable fiction. sparschuh_gothic_style440Hz.scl ! 12 ! 555/526 ! C# 277.5 Hz 285/263 ! D 312/263 ! Eb 329/263 ! E 351/263 ! F 370/263 ! F# 394/263 ! G 416/263 ! G# 440/263 ! A reference-pitch 440Hz 468/263 ! Bb 987/526 ! B 493.5 Hz 2/1 on the keys +----------- | C 263 middle-C +--|277.5=C# | D 285 +--|312=Eb | E 329 +----------- | F 351 +--|370=F# | G 394 +--|416=G# | A 440 +--|468=Bb | B 493.5 +----------- | C'526 &ct. If you dont't like any of that, it's up to you to create yours own personal version, according yours private preferences. A.S.
From: Andreas Sparschuh (2007-04-17) Subject: New septenarius for a'=440 Hz, was Re: Werckmeister's Septinarius temperament --- In [email protected], "Tom Dent" <stringph@...> wrote: > > > 1113/1112 c#"556 c#'278 c#139 > > 417/416 g#208 G#104 > > eb'312 eb156 Eb78 > > bb'468 bb234 Bb117 > > f'351 > > 1053/1052 c"526 c'263/262 131 > > g'393/392 196 98 49=7*7 > > I don't think this is so good, you have C-G tempered by 262/263 > which > is about 1/3 comma... agreed, hence i do return to W's original 131. > > Better to have 315/350 f'350 f175 and 525/524 c''524 c262 131 ... ? > also right, hence so the resulting ratios get even more simple: The calculations benefit from that by less computational0 overhead. Follow the classical way: start traditional @ pitch-class GAMMA=G alike in: http://www.celestialmonochord.org/log/images/celestial_monochord.jpg GG 49 := 7*7 (GAMMA-ut, the empty string in the picture) D 147 3D441 > a'440 a220 A110 AA55Hz=the AA-string of a double-bass e 165 3e495 > b'494 b247 3b741 > f#"740 f#'370 f#185 c# "555 3c#"1665 > g#"'1664 g#"832 g#'416 g#208 G#104 GG#52 GGG#26 GGGG#13 EEb 39 Bb 117 3Bb351 > f'350 f175 3f525 > c"524 c'263 c131 3c393 > g'392 g196 G98 GG49=7^2 returned !septenarius_GG49Hz.scl sparschuh's version @ middle-c'=262Hz or a'=440Hz 12 !absolute pitches relativ to c=131 Hz 555/524 ! c# 138.75 Hz 147/131 ! d 156/131 ! eb 165/131 ! e 175/131 ! f 185/131 ! f# 196/131 ! g 208/131 ! g# 220/131 ! a 440Hz/2 234/131 ! bb 247/131 ! b 2/1 That results on my old piano in the first/lowest octave: AAA 27.5 Hz lowest pitch, on the first white key on the left side BBBb29.25 next upper black key BBB 30.875 http://en.wikipedia.org/wiki/Double_bass "at~30.87 hertz".. CC_ 32.875 CC# 34.6875 := c"555/16 DD_ 36.75 EEb 39 := GGGG#13Hz*3 EE_ 41.25 ..."E1 (on standard four-string basses) at ~41.20 Hz FF_ 43.75 FF# 46.25 GG_ 49 := 7*7 Werckmeister's/Scheibler's initial septimal choice GG# 52 = GGGG#13Hz*4 AA_ 55 = 440Hz/8 ; 3 octaves below Scheibler's choice http://mmd.foxtail.com/Tech/jorgensen.html #133: "Johann Heinrich Scheibler's metronome method of 1836" http://www.41hz.com "41 Hz is the frequency of the low E string on a double bass or an electric bass." if it has none additional 5th string for midi(B0)=BBB 30.875 Hz an 2nd above AAA 27.5 Hz, the lowest A on the piano, without attending or even careing "string-imharmonicty" http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN0000760000S1000S22000004&idtype=cvips&gifs=yes Consider the 3rds qualities in violins empty sting G-D-A-E block order: 1: G > B > Eb> G 2: D > F#> Bb> D 3: A > C#> F > A 4: E > G#> C > E 1: G > B > Eb > G. absolute analysis: GG 49. 5*49 = 245 < b'247 < 248 124 62 31 5*31 = 155 < eb156 Eb78 EEb39 5*39 = 195 < g196 G98 GG7*7. relative diesis 128/125 subpartition: G 123.5/122.5 B 96/95 Eb 196/195 G G ~14.1 cents B ~18.1 Eb ~ 8.86c G 2: D > F# > Bb > D. abs: d147. < 148 74 37 5*37 = f#185 < 186 93 5*31*3=3*155 < 156*3 78*3 39*3 = Bb117 5*39*3=3*195 < 196*3 98*3 49*3 = d147. rel: D 148/147 F# (1/9+85)/(84+1/9) Bb 196/195 D D ~ 11.4c F# ~ 20.5 cents Bb ~ 8.86cents D 3: A > C# > F > A. abs: AA55. A110 < 111 = 37*3 5*111= c#"555 = 5*111 < 112*5 56*5 28*5 14*5 7*5 5*35 = f175 < 176 88 44 22 11 5*11 = AA55. rel: A 111/110 C# 112/111 F 176/175 A ; with all 3 factors superparticular A ~ 15.7c C# ~ 15.5c F ~ 9.86c A 4: E > G#> C > E. abs: e165. < 166 83 5*83 = 415 < g#'416 g#208 G#104 GG#52 GGG#26 GGGG#13 §§ 5*13 = 65 130 < c131 < 132 66 33 5*33 = e165. rel: E (6/7+118)/(117+6/7) G# 131/130 C 132/131 E E ~ 14.6 cents ~ G# ~ 13.3 cents C ~ 13.2c E §§ GGGG# 13 Hz has negative "midi"-index G#_-1, which midi-keyboard supports negative key indices? Summary: 3rds martix in "Cents" 5ths in top>down order 3rds in left>right direction respectively: 1: G 14.1 B_ 18.1 Eb 8.86 G 2: D 11.4 F# 20.5 Bb 8.86 D 3: A 15.7 C# 15.5 F_ 9.86 A 4: E 14.6 G# 13.3 C_ 13.2 E Conversely "ET" detunes all 3rds about the same amount: (128/125)^(1/3) = ~13.7Cents or ~127/126, Attend that: the "septenarius" fits therefore better than ET to horns and trumpets in Eb,Bb & F, with inherent natural 3rds Eb>G, Bb>D, & F>A that turn out less than 10Cents out of tune in the septenarius case. > (cf. 12ET frequencies: 261.6, 277.2, 293.7, 311.1, 329.6, 349.2, > 370.0, 392.0, 415.3, 440.0, 466.2, 493.9 ...) In the "ET" case, it is difficult to resolve the septenarian root-factors above alike: 11,13,31,37 & 83 below the well known 3,5 & 7 limits. > > Can one use the nice ratio 63/50 = 1.26 to build a 'septenarian' >near- > equal tuning? on the one hand is: (63/50)(4/5)=126/125 but but on the other side the diesis 128/125=(128/127)(127/126)(126/125) contains 3 factors. hence: (63/50)^3 = 2.000376... > 2/1 overstretched octave or as superparticular ratio: ((63/50)^3)/2= 250047/250000=(7/47+5320)/(5319+7/47) ~1/2 per mille but a better approximation of the octave delivers 127/126 the factor in the middle: ((5 / 4) * (127 / 126))^3 = ~1.99999802............ > eg Eb-G = 150:189 ... > via Eb150 Bb225 (675) F337 (1011) C505 (504) G378=189 > > then continue: > ... (567) D283 (849) A424=212 (636) E635 (634) B951 (2853/2848) > F#356=178 C#267 (801) G#400 Eb300 > > seems to work nicely at late Baroque pitch levels - only three pure > fifths between Eb-Bb, G#-Eb, F#-C#. > that is expanded: A 424 212 3A636 > E635 > 634 317 B951 > 950 475 3*475=1425 > F#1424 712 356 178 89 C# 267 801 > G# 800 400 200 100 50 25 Werckmeister's tief-Cammerthon 400Hz Eb 75 Bb 225 675 > F 674 337 1011 > C 1050 505 > 504 252 126 63 G 189 567 > D 566 283 849 > A 848 424 recombining that 5ths-circle in ascending ordered pitches yields: C 252.5 Hz middle_C C#267 D 283 Eb300 E 317.5 or better 317? F 337 F#356 G 378 G#400 Beekman's, Descartes's, Mersenne's & Sauveur's standard-pitch A 424 Bb450 B 475.5 or better 475? C'505 T.D. remarked already in his numbers inbetween: >B951 (2853/2848) > F#356 that there appears an unsatisfactory irregular gap: of 2853/2848 = 570.6/569.6 = (951/950)(1425/1424) induced by the choice of E 635 > 634 317 instead 635 > E 634 317 That results in the above none-integral superparticular ratio bug. Hence i do suggest to replace by the tiny changes 1: E 635-->>>634 2: B 951-->>>950 in order to fix the bug. so that now all 5th-tempering steps become integral superparticular ratios, without any exception: A 424 212 106 3A=318 E 317 instead formerly 317.5 3E=951 B 950 475 instead formerly 951 475.5 3B=1425 F# 1424 712 356 178 89 &ct. the rest of the circle remains unchanged. Is that ok? Analysis of the: 3rds sharpness, -how much wider than 5/4- per diesis subpartition into superparticular factors, so that the product of 3 tempered 5ths results an octave in each of the 4 blocks: 1: G > B > Eb > G. abs: G378. 189 > 190 95 5*95 = B 475 < 480 240 120 60 30 15 5*15 = Eb75 5*75 = 3*125 < 126*3 = G378. rel. 2^7/5^3=128/125= G 190/189 B 160/159 Eb 126/125 G remember (63/50)(4/5)=126/125 ?or formerly in the original version: ?G378. 189 > 190 95 ?5*95 = 475 950 < B951 < 960 480 240 120 60 30 15 ?...... ?rel. 2^7/5^3=128/125= ?G (190/189)(951/950) B 320/317=(2/3+106)/(105+2/3) Eb 126/125 G That appears i.m.o. much more complicated than my suggested change. 2: D > F# > Bb > D. abs: D283. < 284 142 71 5* 71 = 355 < F#356 178 89 < 90 45 5* 45 = Bb225 < 226 113 5*113 = 565 < D556 283. rel. 128/125= D (284/283)(356/355) F# 90/89 Bb (226/225)(556/565) D 3: A > C# > F > A. abs: A424. 212 106 53 5* 53 = 265 < C#267 < 268 134 67 5* 67 = 335 < F 337 < 338 169 5*169 = 845 < A 848 424. rel: 128/125= A 133.5/123.5 C# (268/267)(168.5/166.5) F >>> >>> F (338/337)((2/3+282)/(281+2/3)) A 4: E > G# > C > E. E 317. < 320 160 80 40 20 5* 20 = G#100 < 101 5*101 = C 505 < 506 253 5*253 = 1265 < E 1268 634 317. rel: 128/125= E (2/3+106)/(105+2/3) G# 101/100 C (506/505)/(422.666.../421.666...) E ?or formerly ?E 635? > 640 320 ..... ?....&ct. alike above... ?.... ?5*253 =1265 < 1270 635? try to find out similar improvements in order to reduce the ratios to less complicated proportions have a lot of fun in whatever tuning you do prefer A.S.